Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2011, Article ID 479576, 14 pages
doi:10.1155/2011/479576
Research Article
Hypersingular Marcinkiewicz Integrals along
Surface with Variable Kernels on Sobolev Space
and Hardy-Sobolev Space
Wei Ruiying and Li Yin
School of Mathematics and Information Science, Shaoguan University, Shaoguan 512005, China
Correspondence should be addressed to Wei Ruiying, weiruiying521@163.com
Received 30 June 2010; Revised 5 December 2010; Accepted 20 January 2011
Academic Editor: Andrei Volodin
Copyright q 2011 W. Ruiying and L. Yin. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Let α
≥
0, the authors introduce in this paper a class of the hypersingular
Marcinkiewicz integrals along surface with variable kernels defined by μΦ
Ω,α f x
∞ 1/2
2
n−1
32α
∞
n
q
n−1
0 | |y|≤t Ωx, y/|y| f x − Φ|y|y dy| dt/t
, where Ωx, z ∈ L Ê × L Ë with q > max{1, 2n − 1/n 2α}. The authors prove that the operator μΦ
Ω,α is bounded from
p
p
n
p
n
Sobolev space Lα Ê to L Ê space for 1 < p ≤ 2, and from Hardy-Sobolev space Hα Ên to
p
n
2
n
L Ê space for n/n α < p ≤ 1. As corollaries of the result, they also prove the L̇α R − L2 Rn boundedness of the Littlewood-Paley type operators μΦ
and μ∗,Φ
which relate to the Lusin
Ω,α,λ
Ω,α,S
∗
area integral and the Littlewood-Paley gλ function.
1. Introduction
Let Ên n ≥ 2 be the n-dimensional Euclidean space and Ën−1 be the unit sphere in Ên
equipped with the normalized Lebesgue measure dσ dσ·. For x ∈ Ên \ {0}, let x x/|x|.
Before stating our theorems, we first introduce some definitions about the variable
kernel Ωx, z. A function Ωx, z defined on Ên × Ên is said to be in L∞ Ên × Lq Ën−1, q ≥ 1,
if Ωx, z satisfies the following two conditions:
1 Ωx, λz Ωx, z, for any x, z ∈ Ên and any λ > 0;
1/q
2 ΩL∞ Ên×Lq Ën−1 supr≥0, y∈Ên Ën−1 |Ωrz y, z |q dσz < ∞.
In 1955, Calderón and Zygmund 1 investigated the Lp boundedness of the singular
integrals TΩ with variable kernel. They found that these operators connect closely with the
2
Journal of Inequalities and Applications
problem about the second-order linear elliptic equations with variable coefficients. In 2002,
Tang and Yang 2 gave Lp boundedness of the singular integrals with variable kernels
associated to surfaces of the form {x Φ|y|y }, where y y/|y| for any y ∈ Ên \ {0} n ≥
2. That is, they considered the variable Calderón-Zygmund singular integral operator TΩΦ
defined by
TΩΦ f x
Ω x, y p·v·
n f x − Φ y y dy.
Ên y 1.1
On the other hand, as a related vector-valued singular integral with variable kernel,
the Marcinkiewicz singular with rough variable kernel associated with surfaces of the form
{x Φ|y|y } is considered. It is defined by
μΦ
Ω f x ∞ 2 dt 1/2
Φ
,
FΩ,t x 3
t
0
1.2
where
Φ
FΩ,t
x
|y|≤t
Ë
n−1
Ω x, y n−1 f x − Φ y y dy,
y Ω x, z dσ z 0.
1.3
1.4
If Φ|y| |y|, we put μΦ
Ω μΩ . Historically, the higher dimension Marcinkiewicz
integral operator μΩ with convolution kernel, that is Ωx, z Ωz, was first defined and
studied by Stein 3 in 1958. See also 4–6 for some further works on μΩ with convolution
kernel. Recently, Xue and Yabuta 7 studied the L2 boundedness of the operator μΦ
Ω with
variable kernel.
Theorem 1.1 see 7. Suppose that Ωx, y is positively homogeneous in y of degree 0, and satisfies
1.4 and
1/q
2 supy∈Ên Ën−1 |Ωy, z |q dσz < ∞, for some q > 2n − 1/n. Let Φ be a positive
and monotonic (or negative and monotonic) C1 function on 0, ∞ and let it satisfy the
following conditions:
i δ ≤ |Φt/tΦ t| ≤ M for some 0 < δ ≤ M < ∞;
ii Φ t is monotonic on 0, ∞.
Then there is a constant C such that μΦ
Ω f2 ≤ Cf2 , where constant C is independent of f.
Since the condition 2 implies 2 , so the L2 Ên boundedness of μΦ
Ω holds if Ω ∈
L Ê × Lq Ën−1 with q > 2n − 1/n.
Our aim of this paper is to study the hypersingular Marcinkiewicz integral μΦ
Ω,α along
surfaces with variable kernel Ω, and with index α ≥ 0, on the homogeneous Sobolev space
∞
n
Journal of Inequalities and Applications
3
Lα Ên for 1 < p ≤ 2 and the homogeneous Hardy-Sobolev space Hα Ên for some n/nα <
Φ
x be as above, we then define the operators μΦ
p ≤ 1. Let FΩ,t
Ω,α by
p
p
μΦ
Ω,α f x ∞ 2 dt 1/2
Φ
,
FΩ,t x 32α
t
0
α ≥ 0.
1.5
Our main results are as follows.
Theorem 1.2. Suppose that α ≥ 0, Ωx, y satisfies 1.4 and Ω ∈ L∞ Ên × Lq Ën−1 with q >
max{1, 2n − 1/n 2α}. Let Φ be a positive and increasing C1 function on 0, ∞ and let it satisfy
the following conditions:
i Φt tΦ t;
ii 0 ≤ Φ t ≤ W on 0, ∞.
Then there is a constant C such that μΦ
Ω,α fL2 Ên ≤ CfL2α Ên , where constant C is independent
of f.
Theorem 1.3. Suppose 0 < α < n/2, and that Ω ∈ L∞ Ên × Lq Sn−1 , with q > max{1, 2n −
1/n 2α}, and satisfies 1.4. Let Φ be a positive and increasing C1 function on 0, ∞ and let it
satisfy the following conditions:
i Φt tΦ t;
ii 0 < Φ t ≤ 1, Φ0 0.
p
Then, for n/n α < p ≤ 1, there is a constant C such that μΦ
Ω,α fLp Ên ≤ CfHα Ên ,
where constant C is independent of any f ∈ Hα Ên ∩ SÊn .
p
Furthermore, our result can be extended to the Littlewood-Paley type operators μΦ
Ω,α,S
μ∗,Φ
Ω,α,λ
and
with variable kernels and index α ≥ 0, which relate to the Lusin area integral and
Φ
x be as above, we then define the
the Littlewood-Paley gλ∗ function, respectively. Let FΩ,t
∗,Φ
n
operators μΦ
Ω,α,S and μΩ,α,λ for f ∈ SÊ , respectively by
μΦ
Ω,α,S f x
⎛
∗,Φ μΩ,α,λ f x ⎝
with λ > 1, where Γx {y, t ∈
have the following conclusion.
Φ 2 dydt
FΩ,t y n32α
t
Γx
λn
1/2
,
⎞1/2
Φ 2 dydt ⎠
,
FΩ,t y n32α
t
1.6
Ê
t
t x − y Ên1
: |x − y| < t}. As an application of Theorem 1.2, we
n1
Theorem 1.4. Under the assumption of Theorem 1.2, then Theorem 1.2 still holds for μΦ
Ω,α,S and
μ∗,Φ
Ω,α,λ .
By Theorems 1.2 and 1.3 and applying the interpolation theorem of sublinear operator,
p
we obtain the Lα − Lp boundedness of μΦ
Ω,α .
4
Journal of Inequalities and Applications
Corollary 1.5. Suppose 0 < α < n/2, and that Ω ∈ L∞ Ên × Lq Sn−1 , q > max{1, 2n − 1/n 2α}, and satisfies 1.4. Let Φ be given as in Theorem 1.3. Then, for 1 < p ≤ 2, there exists an absolute
positive constant C such that
Φ μΩ,α f Ê
Lp n
≤ Cf Lpα Ên ,
1.7
for all f ∈ Lα Ên ∩ SÊn .
p
Remark 1.6. It is obvious that the conclusions of Theorem 1.2 are the substantial improvements and extensions of Stein’s results in 3 about the Marcinkiewicz integral μΩ with
convolution kernel, and of Ding’s results in 8 about the Marcinkiewicz integral μΩ with
variable kernels.
Remark 1.7. Recently, the authors in 9 proved the boundedness of hypersingular
p
Marcinkiewicz integral with variable kernels on homogeneous Sobolev space Lα Rn for
1 < p ≤ 2 and 0 < α < 1 without any smoothness on Ω. So Corollary 1.5 extended the results
in 9, Theorem 5.
Throughout this paper, the letter C always remains to denote a positive constant not
necessarily the same at each occurrence.
2. The Bounedness on Sobolev Spaces
Before giving the definition of the Sobolev space, let us first recall the Triebel-Lizorkin space.
Fix a radial function ϕx ∈ C∞ satisfying suppϕ ⊆ {x : 1/2 < |x| ≤ 2} and 0 ≤
ϕx ≤ 1, and ϕx > c > 0 if 3/5 ≤ |x| ≤ 5/3. Let ϕj x ϕ2j x. Define the function ψj x by
Fψj ξ ϕj ξ, such that Fψj ∗ fξ Ffξϕj ξ.
α,q
For 0 < p, q < ∞, and α ∈ Ê, the homogeneous Triebel-Lizorkin space Ḟp is the set of
all distributions f satisfying
⎫
⎧
1/q ⎪
⎪
⎬
⎨
q
α,q
n
n
−αk
Ḟp Ê f ∈ S Ê : f Ḟpα,q <∞ .
2 ψk ∗ f ⎪
⎪
k
⎭
⎩
2.1
p
For p ≥ 1, the homogeneous Sobolev spaces Lα Ên is defined by Lα Ên Ḟpα,2 Ên ,
namely fLpα fḞpα,2 . From 10 we know that for any f ∈ L2α Ên p
f ∼
L2α Ên Ên
p
2 2α
F f ξ |ξ| dξ
1/2
,
2.2
and if α is a nonnegative integer, then for any f ∈ Lα Ên p
f p
∼
Ên Lα D τ f |τ|α
Ên .
Lp 2.3
Journal of Inequalities and Applications
5
For 0 < p ≤ 1, we define the homogeneous Hardy-Sobolev space Hα Ên by Hα Ên Ê It is well known that H p Ên Ḟp0,2 Ên for 0 < p ≤ 1, one can refer 10 for the
details.
Next, let us give the main lemmas we will use in proving theorems.
p
p
Ḟpα,2 n .
Lemma 2.1 see 11. Suppose that n ≥ 2 and f ∈ L1 Ên ∩ L2 Ên has the form fx f0 |x|P x where P x is a solid spherical harmonic polynomial of degree m. Then the Fourier
transform of f has the form Ffx F0 |x|P x, where
F0 r 2πi−m r −n2m−2/2
∞
f0 sJn2m−2/2 2πrssn2m/2 ds,
2.4
0
and r |ξ|, Jm s is the Bessel function.
Lemma 2.2 see 12. For λ n − 2/2, and −λ ≤ α ≤ 1, there exists C > 0 such that for any
h ≥ 0 and m 1, 2, . . .,
h J t C
mλ
dt ≤ λα .
m
0 tλα
2.5
Lemma 2.3. Let α ≥ 0, λ n − 2/2, Φ is a C1 function on 0, ∞ and let it satisfy the conditions
(i) and (ii) in Theorem 1.2.
∞
1/2
Denote gα fx 0 |Nε fx|2 dε/ε12α , if
F Nε f ξ Φε|ξ|
0
Jmλ t
dt · F f ξ.
tλ1
2.6
Then there exists a constant C independent of m, such that gα fL2 ≤ C/mλ1α fL2α for every
integer m ∈ Æ , m > α.
Proof. Let η|x| |x|
0
Jmλ t/tλ1 dt, then we have
2
g α f 2
Ên
∞
0
∞ Nε fx2 dε dx
ε12α
ηΦε|ξ|F f ξ2 dξ dε
ε12α
0
Ê
∞ 2
η Φ β |ξ| F f ξ |ξ|2α dξ dβ
|ξ|
β12α
0
Ên
∞ 2
η Φ β |ξ| dβ F f ξ2 |ξ|2α dξ.
12α
|ξ|
β
Ên 0
n
∞
2
ηΦβ/|ξ||ξ|2 dβ/β12α ≤ C/mλ1α .
∞ m/2 ∞
Decompose this integral into two parts 0 0 m/2 : I1 I2 .
So it suffices to show
0
2.7
6
Journal of Inequalities and Applications
For I2 , by using Lemma 2.2 and Φt tΦ t, we can get
I2 ∞ Φβ/|ξ||ξ|
m/2
≤
≤
0
C
m2λ2
∞
m/2
Jmλ t
dt
tλ1
2
dβ
β12α
dβ
β12α
2.8
C
.
m2λ22α
For the other part I1 , applying Stirling’s formula, we have
√
√
2πxx−1/2 e−x ≤ Γx ≤ 2 2πxx−1/2 e−x .
2.9
Also in 13, the authors proved the following inequality
|Jν t| ≤
t/2ν
.
Γν 1
2.10
So by 2.9 and 2.10, 0 ≤ α < α 1 ≤ m, and noting that Φt ≤ Wt, we have
I1 m/2 Φβ/|ξ||ξ|
0
≤
0
m/2 Φβ/|ξ||ξ|
0
0
Jmλ t
dt
tλ1
2
|Jmλ t|
dt
tλ1
1
≤ 2m2λ 2
2
Γ m λ 1
dβ
β12α
2
dβ
β12α
m/2 Φβ/|ξ||ξ|
0
0
tmλ
dt
tλ1
2
dβ
β12α
m/2 2m
dβ
β
Φ
|ξ|
β12α
0
m/2 2m dβ
β
e2m2λ2
Φ
≤
12α−2m
2m2λ1
|ξ|
β
2π22m2λ m λ 1
0
≤
1
22m2λ Γ2 m λ 1
≤C
≤C
≤
m/2
e2m2λ2
2π22m2λ m
e 2m2λ2
4
2m2λ1
λ 1
0
dβ
β12α−2m
1
m2α2λ2
C
.
m2α2λ2
So far we can deduce the desired conclusion of Lemma 2.3.
2.11
Journal of Inequalities and Applications
7
Proof of Theorem 1.2. The basic idea of proof can go back to 14, for recently papers, one see 8,
15. By the same argument as in 1, let {Ym,j } m ≥ 1, j 1, 2, . . . , Dm denote the complete
system of normalized surface spherical harmonics. See 14 for instance, we can decompose
Ωx, y as following:
Dm
∞ Ω x, y am,j xYm,j y is a finite sum.
2.12
m1 j1
Denote
⎞1/2
⎛
Dm 2
am x ⎝ am,j x ⎠ ,
bm,j x j1
am,j x
,
am x
2.13
then we get
Dm
2
bm,j
x 1,
Dm
∞
am x bm,j xYm,j y .
Ω x, y j1
m1
2.14
j1
Then, applying Hölder inequality twice, we have for any 0 < ε < 1 that
2
∞
2 ∞ Ym,j y dt
Φ
bm,j x n−1 f x − Φ y y dy 32α
μΩ,α fx t
y
0 |y|≤t m1
≤
∞
a2m xm−ε12α
m1
∞
mε12α
m1
2
∞ Dm
Ym,j y dt
×
b
f
x
−
Φ
y
dy
y
x
m,j
n−1
t32α
y 0 |y|≤t j1
≤
∞
a2m xm−ε12α
m1
∞
mε12α
m1
∞
0
⎛
⎝
Dm
⎞
2
bm,j
x⎠
j1
2
Dm Ym,j y dt
f x − Φ y y dy 32α
× t
|y|≤t yn−1
j1
∞
a2m xm−ε12α
m1
∞
mε12α
m1
2
∞ Dm Ym,j y dt
×
f x − Φ y y dy 32α .
t
|y|≤t yn−1
0
j1
2.15
8
Journal of Inequalities and Applications
By 14, page 230, equation 4.4, we can observe that the series in the first parenthesis
on the right-hand side of the inequality above, for each x fixed, is equal to Ωx, ·2L2 Ën−1 ,
−γ
where L2−γ Ën−1 is the Sobolev space on Ën−1 with γ ε1/2 α for any 0 < ε < 1. So if we
take ε sufficiently close to 1, then by the Sobolev imbedding theorem Lq ⊂ L2−γ , we have
1/2
2
−ε12α
am xm
≤ CΩL∞ Ên×Lq Ën−1 : CΩ
2.16
m
with q > max{1, 2n − 1/n 2α}.
By Fourier transform and 2.16, we get
2
Dm Ym,j y dt
n−1 f x − Φ y y dy dx 32α
t
n
0
Ê j1 |y|≤t y
m1
2
Dm ∞ ∞
y
Y
m,j
2
y y dy ξ dξ dt
≤ CΩ mε12α
f
·
−
Φ
F
n−1
t32α
Ên |y|≤t y m1
j1 0
∞ ∞
Φ 2
μΩ,α f ≤ CΩ2 mε12α
2
∞
: CΩ2
mε12α
m1
Dm Φ 2
μΩ,j,α f .
j1
2
2.17
For μΦ
Ω,j,α f, we have
2
Ym,j y −2πix·ξ
dt
y
e
f
x
−
Φ
y
dx
dy
dξ 32α
n−1
t
0
Ên |y|≤t Ên y
∞ Ym,j y −2πiΦ|y|y ·ξ
e
0
Ên |y|≤t yn−1
2 ∞ Φ
μΩ,j,α f 2
2
dt
f x − Φ y y e−2πix−Φ|y|y ·ξ dx dy dξ 32α
t
n
Ê
2
∞ Ym,j y −2πiΦ|y|y ·ξ 2
dt
dy F f ξ dξ 32α
n−1 e
t
n
0
Ê |y|≤t y
2
∞ Ym,j y −2πiΦ|y|y ·ξ dt 2
1
F f ξ dξ.
e
dy
t
Ên 0 t1α |y|≤t yn−1
×
2.18
Journal of Inequalities and Applications
9
For the integral on the right hand side of the above inequality, by changing of variable, we
can get
1
t1α
|y|≤t
Ym,j y −2πiΦ|y|y ·ξ
dy
n−1 e
y t 1
t1α
Ën−1
0
Ym,j y e−2πiΦsy ·ξ dy ds
Φt 1
t1α
Ën−1
0
1
t1α
|y|≤Φt
Ym,j y e−2πiγ y ·ξ Φ−1 γ dy dγ
2.19
Ym,j y −2πiy·ξ −1 Φ y dy.
n−1 e
y
So we have
2 Φ
μΩ,j,α f 2
2
∞ Ym,j y −2πiy·ξ −1 dt 2
1
F f ξ dξ.
y
Φ
e
dy
t
Ên 0 t1α |y|≤Φt yn−1
2.20
Φ,m,j
Put Pm,j x Ym,j x |x|m and ϕt,α
we can deduce from Lemma 2.1 that
x Pm,j x · |x|−n−m1 χ|x|≤Φt xΦ−1 |x| t−1−α ,
Φ,m,j F ϕt,α
ξ Pm,j |ξ| · F0 |ξ| Ym,j ξ · |ξ|m F0 |ξ|,
2.21
where
−m −n/2−m1
F0 r 2πi
Φt
r
t−1−α s−n−m1 Φ−1 s Jn/2m−1 2πrssn/2m ds
0
−m −n/2−m1 −1−α
2πi
r
Φt
t
0
s−n/21 Jn/2m−1 2πrsd Φ−1 s
Jn/2m−1 2πrΦ β
2πi r
t
dβ
n/2−1
0
Φ β
−α t J
n/2m−1 2πrΦ β
n/2 −m −m t
i r
dβ.
2π
t 0 2πrΦβn/2−1
−m −n/2−m1 −1−α
t
2.22
10
Journal of Inequalities and Applications
Hence, we have
Φ 2
μΩ,j,α f ∞ 0
∞ 2
2 dt
Φ,m,j
ϕt,α ∗ fx dx
t
Ên
2 dt
Φ,m,j
F ϕt,α ∗ f ξ dξ
t
0
Ên
2
∞ −α t J
m −m −m
n/2m−1 2π|ξ|Φ β
dt 2
n/2 t
≤
F f ξ dξ
dβ
Ym,j ξ |ξ| i |ξ| 2π
n/2−1
t
t 0
Ên 0 2π|ξ|Φ β
2
∞ 1 t Jn/2m−1 2π|ξ|Φ β
dt 2
≤C
dβ
Ym,j ξ
12α F f ξ dξ.
n/2−1
t
t 0
Ên 0 2π|ξ|Φ β
2.23
By 14, we know that
So we can get
Dm 2
Φ
μΩ,j,α f ≤ Cmn−2
j1
2
Dm
j1
|Ym,j z |2 ∼
mn−2 .
2
∞ t
Jn/2m−1 2π|ξ|Φ β
dt 2
1
dβ 12α F f ξ dξ.
n/2−1
t
Ên 0 t 0 2π|ξ|Φ β
2.24
Set λ n/2 − 1, ρ 2π|ξ|Φβ and note that Φt tΦ t, we can deduce that
Jn/2m−1 2π|ξ|Φ β
n/2−1 dβ
0
2π|ξ|Φ β
1 2π|ξ|Φt Jmλ ρ
1
dρ
t 0
ρλ
2π|ξ|Φ Φ−1 ρ/2π|ξ|
ρ
1 2π|ξ|Φt Jmλ ρ −1
dρ.
Φ
t 0
2π|ξ|
ρλ1
1
U :
t
t
2.25
Noting that Φt is increasing, by using the second mean-value theorem, we get, for
some 0 ≤ η < 2π|ξ|Φt,
2π|ξ|Φt
1
ρ
J
mλ
dρ
|U| ≤ Φ−1 Φt
λ1
t
ρ
η
2π|ξ|Φt Jmλ ρ ≤
dρ.
0
ρλ1
2.26
Journal of Inequalities and Applications
11
From 2.26, it follows that
Dm Φ 2
n−2
μm,j,α f ≤ Cm
2
j1
2
∞ 2π|ξ|Φt
dt
Jmλ ρ
dρ · F f ξ 12α dξ.
t
ρλ1
Ên 0 0
2.27
Thus using Lemma 2.3, we can deduce the desired conclusion of Theorem 1.2.
λn ∗,Φ
Proof of Theorem 1.4. First, we know that μΦ
Ω,α,S fx ≤ 2 μΩ,α,λ fx. On the other hand,
2
∗,Φ
μΩ,α,λ f
2
2
λn dzdt
1
Ωx, z dz
f
x
−
Φ|z|z
n12α dx
n−1
t
n1
t
n
Ê
R
|z|≤t |z|
⎛
2
λn ⎞ ∞ dzdt
1
t
Ωx, z 1
⎠
⎝
dz
dx
f
x
−
Φ|z|z
12α
n
n−1
t
t
t
n
n
0
Ê
Ê t x−y
|z|≤t |z|
t
t x − y
2
≤ CμΦ
Ω,α f .
2
2.28
Thus, using Theorem 1.2, we can finish Theorem 1.4.
3. The Bounedness on Hardy-Sobolev Spaces
In order to prove the boundedness for operator μΦ
Ω,α on Hardy-Sobolev spaces and prove
Theorem 1.3, we first introduce a new kind of atomic decomposition for Hardy-Sobolev space
as following which will be used next.
Definition 3.1 see 16. For α ≥ 0, the function ax is called a p, 2, α atom if it satisfies the
following three conditions:
1 suppa ⊂ B with a ball B ⊂ Ên ;
2 aL2α ≤ |B|1/2−1/p ;
3 Ên axP x 0, for any polynomial P x of degree ≤ N n1/p − 1α.
By 16, we have that every f ∈ Hα Ên can be written as a sum of p, 2, α atoms
aj x, that is,
p
f
λj aj .
j
3.1
12
Journal of Inequalities and Applications
Proof of Theorem 1.3. Similar to the argument of Lemma 3.3 in 17 and using above atomic
decomposition, it suffices to show that
p
Φ
μΩ,α a p ≤ C,
3.2
L
with the constant C independent of any p, 2, α atom a.
Assume suppa ⊂ B0, R. We first note that
p
Φ
μΩ,α a p ≤
L
|x|≤8R
p
Φ
μΩ,α ax dx |x|>8R
p
Φ
μΩ,α ax dx
3.3
: U1 U2 .
For U1 , using Theorem 1.2, it is not difficult to deduce that
p
p
n1−p/2
U1 ≤ CμΦ
≤ CaL2 Rn1−p/2
a
2R
Ω,α
L
α
3.4
≤ CRnp/2−1 Rn1−p/2 ≤ C.
For U2 , we first consider the case n/n α < p < 1, according to 15, Lemma 5.5, for
0 < α < n/2 and p, 2, α atom a with support B B0, R, one has
|ax|dx ≤ CRn−n/pα .
3.5
B
Using Minkowski inequality and Hölder inequality for integrals, and 3.5, we can get
U2 |x|>8R
p
Φ
μΩ,α ax dx
⎞p/2
2
∞ dt
Ω x, y ⎝
n−1 a x − Φ y y dy 32α ⎠ dx
t
0 |y|≤t y |x|>8R
⎛
p
Ω x, y ≤
nα a x − Φ y y dy dx.
y
|x|>8R Ên
3.6
Journal of Inequalities and Applications
13
For the integral on the right hand side of the above inequality, by changing of variable and
noting that 0 < Φ t ≤ 1, Φ0 0, we can get
Ω x, y nα a x − Φ y y dy
Ên y
R Ω x, y a x − Φry dr dy
1α
r
Ën−1 0
ΦR Ω x, y a x − γ y 1 dγ dy
1α
Φ Φ−1 γ
Ën−1 0
Φ−1 γ
ΦR Φ−1 γ
Ω x, y dγ dy
−1 1α a x − γ y
γ
Ën−1 0
Φ γ
ΦR Ω x, y −1 α a x − γ y dγ dy
Φ γ
γ
Ën−1 0
Ω x, y n α a x − y dy
Φ−1 y
|y|≤ΦR y
Ω x, x − y α a y dy
n −1 x − y |x−y|≤ΦR x − y Φ
Ω x, x − y a y dy.
≤
x − ynα
|x−y|≤ΦR
3.7
By 3.7, we can get
U2 ≤
Ω x, x − y p
a y dy dx
x − ynα
2j R<|x|<2j1 R Ên
∞ j3
p
|Ωx, x − y| ≤
nα a y dy dx
2j R<|x|<2j1 R Ên x − y j3
p
∞ n1−p Ω x, x − y j
a y
≤
dx dy
2R
x − ynα
B
2j R<|x|<2j1 R
j3
∞ n1−p 2R
j
p
≤ CΩL∞ ×L1
a y dy
3.8
p ∞ −αp n1−p
2j R
2j R
·
.
B
j3
Thus by 3.5 and the condition p > n/n α,
p
U2 ≤ CΩL∞ ×L1
∞
j3
2jn−np−αp ≤ C.
3.9
14
Journal of Inequalities and Applications
As for p 1, similar to the argument of n/n α < p < 1, we can easily get U2 ≤ C. So
far the proof of Theorem 1.3 has been finished.
Acknowledgments
This project supported by the National Natural Science Foundation of China under Grant no.
10747141, Zhejiang Provincial National Natural Science Foundation of China under Grant no.
Y604056, and Science Foundation of Shaoguan University under Grant no. 200915001.
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